Can you play Evelyn from face down hidden on the same turn you play her?

No. You cannot play Evelynn from facedown on the same turn you hide her. Hidden cards can only be played from facedown beginning on the next turn (FAQs #3046, #3982, #8114; Rule 811.1.b).

If you want to use Evelynn's "when you play me from face down" ability on your turn, you would need to have hidden her on a previous turn. Alternatively, you can play Evelynn from your hand for her full cost immediately — you don't have to hide her at all.